International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
GRAPHICAL SCHEME FOR DETERMINATION
OF MARKET CLEARING PRICE USING
QUADRATIC BID FUNCTIONS
Tanay Joshi and Gagan Uberoi
School Of Electrical Engineering, VIT University, Vellore 632014
[email protected]
[email protected]
ABSTRACT
Market clearing price (mcp) calculation is one of the most important functions of a power pool operator. Many
existing power pools use a linear bid function received from the generators and the consumers for the
computation of market clearing price. Although quadratic bid functions are more informative and accurate, they
are not generally used, as the process of computing the market clearing price becomes more complicated. In this
paper the final market clearing price and the schedules of available generators and loads are calculated using
quadratic bid functions with the help of a simple iterative scheme coded in MATLAB.
KEYWORDS
Market clearing price (mcp), bid functions, economy of scale, social welfare function, incremental cost.
1. INTRODUCTION
One of the most important factors for the computation of the market clearing price in a deregulated
market is the format of the bids and the cost function. There are four types of cost functions: linear,
quadratic, translog and loglog. Most of the electricity market literatures use a piecewise linear or
quadratic functions to compute the market clearing price. The effect of choosing different type of bids
in electricity auctions and the difference between continuous and discrete bids are discussed by N.
Fabra et al. [1]. According to economy of scale a firm in a competitive market maximises its profit
when the sale is at marginal cost. But the generating companies have to submit the bids according to
the electricity market. The piecewise linear bids are suitable for small or compact markets but for
generating companies with multiple units, there would be a sudden change in cost when one unit is
turned on or off. This jump would make calculus analysis difficult and would thus result in
inconsistencies. Quadratic curves result in smooth revenue, profit curves when constraints like
transmission congestion and generator capacities are not active.
The aim of this paper is to develop a method for determining the market clearing price in the case of
pool markets neglecting the effect of congestion. In order to do so, bids from both the generating side
and consumer side have been considered. Both these bids are considered to be quadratic functions of
Real Power instead of linear functions for an accurate approach [2], [3], [4],[5]. Different cases
considering violation of maximum and minimum limits of individual generators and loads have been
discussed as well. The proposed method has been developed as an extension of an earlier method [2]
used for determining mcp. Extending the concept developed in [2] a graphical approach has been
developed in order to determine final clearing price. The results obtained are in concurrence with the
results as given by [2]. Section 2 discusses the algorithm developed using the formulae for
incremental cost as derived by [2], in a minimum cost approach [6] so as to maximize the social
welfare function in an elastic demand market. A detailed illustration of algorithm is given thereby.
Section 3 gives a detail look at various cases and the comparative results. The final illustration
discusses a 30 bus system with a large number of generators and static as well as dynamic loads.
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Vol. 1,Issue 2,pp.144-150
International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
2. FORMULAE AND ALGORITHM
The objective of a power pool operator is to maximise the social welfare function which is defined as
the difference between the cumulative consumer and generator bids. In the following subsections we
have discussed the formulae used in the basic closed loop solution followed by the algorithm used to
determine the final clearing price and the optimal schedules of generators and loads present in the
system.
2.1 Formulae used in the closed loop solution
The following formulae used for the calculation of the clearing price have been discussed in detail by
Bijuna Kunju K∗ and P S Nagendra Rao [2] and the proposed algorithm is an extension of their work.
Consider a system with N generators and M consumers. Let the generator bid function for the ith
generator be
Ci(Pgi) = aiPg2i + biPgi + ci
(1)
and the consumer benefit function for the jth load be
Bfj(Pdj) = αjPd2j + βjPdj + γj
(2)
Now, the objective of the pool market operator is to maximize the social welfare function
M
N
∑ Bfj(Pdj) - ∑ Ci(Pgi)
j=1
(3)
i=1
Subject to the power balance constraint
N
M
∑ (Pgi) = ∑ (Pdj)
i=1
(4)
j=1
The schedules for each of the generators and the demand of each consumer that can be met is obtained
as,
Pgi = λ-bi
(5)
2ai
Pdi = λ-βj
(6)
2αj
2.2 Flow chart to determine final schedules and clearing price
The given formulae are valid only if the generators and consumers are within their specified limits. In
case of a violation the violating generator’s or load’s schedule is held constant at its limit and the
iterative process is repeated till we get an optimal solution for the clearing price and schedules of each
generator or load. These steps have been implemented in form of a flow chart as shown in figure 1.
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Vol. 1,Issue 2,pp.144-150
International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
Figure 1: Flow Chart to calculate schedules and clearing price
3. ILLUSTRATION OF THE METHOD
Consider a system consisting of three generators and two consumers with variable demands. The
generator cost functions of the three units are,
C1 = 0.003Pg21 + 2Pg1 + 80 Rs/h,
C2 = 0.015Pg22 + 1.45Pg2 + 100 Rs/h,
C3 = 0.01Pg23 + 0.95Pg3 + 120 Rs/h.
The offer functions of the two consumers are,
Bf1 = −0.002Pd21 + 5Pd1 + 150 Rs/h,
Bf2 = −0.001Pd22 + 6Pd2 + 200 Rs/h.
A. Case 1: No Limit Constraints
This case discusses the scheduling of the three generators and two loads when they operate without
any constraints. The mcp obtained in this case is the intersection point of the aggregate supply and
demand curves as shown in figure 2.
Figure 2. Demand and Generation vs Incremental Cost
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Vol. 1,Issue 2,pp.144-150
International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
The final clearing price is : λ = 4.679 Rs/MW
The schedules obtained are: Pg1 = 446.5MW, Pg2 =107.64MW, Pg3 =186.46MW, Pd1=80.2MW, Pd2
=660.4MW
B. Case 2: Maximum Limit on Generator Output
This case discusses the effect of constraining a generator to a certain maximum limit on the clearing
price and the individual schedules of the generators and loads.
In addition to the conditions in case 1, suppose that generator 1 has a maximum limit of 400
MW. Hence, the generation of this unit must be constrained to 400 MW, instead of 446.5 MW as
calculated above in the previous case. The new clearing price obtained is shown in figure 3.
Figure 3. Demand and Generation vs Incremental Cost
The final clearing price is : λ = 4.735 Rs/MW
The schedules obtained after accounting for the violation in generator limits are:
Pg1 = 400.0 MW, Pg2 =109.5 MW, Pg3 =189.25 MW
Pd1 = 66.25 MW, Pd2 =632.5MW
C. Case 3: Maximum Limit on Load
In another case one of the loads is constrained to a maximum limit of 600MW. This implies a
constraint violation as seen by the schedules of case 1 and the load must be fixed at 600 MW
instead of 660.4 MW. The new clearing price obtained after adjusting the schedules is shown
in figure 4.
Figure 4.Demand and Generation vs Incremental Cost
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Vol. 1,Issue 2,pp.144-150
International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
The final clearing price is : λ = 4.558 Rs/MW
The schedules obtained after accounting for the violation in load limits are:
Pg1 = 426.33 MW, Pg2 =103.6 MW, Pg3 =180.4MW
Pd1 =110.5 MW, Pd2 =600.0 MW
D. Case 4: Maximum Limits on Both Generators and Consumer Loads
This case discusses the condition when both generator and load limit gets violated. Generator1 has a
maximum limit of 400MW and the maximum demand of consumer 2 is 600 MW. From the optimal
schedule of case 1 it is seen that both limits get violated. Hence their schedules are readjusted to their
respective limits and a new clearing price is calculated as shown in figure 5.
Figure 5. Demand and Generation vs. Incremental Cost
The final clearing price is : λ = 4.637 Rs/MW
The schedules obtained after accounting for the violation in generator and load limits are:
Pg1 = 400.00MW, Pg2 =106.233MW, Pg3 =184.35MW
Pd1 = 90.75 MW, Pd2 =600MW
E. Case 5: Mixed Load - (No Limit violation)
This example is provided to illustrate the method when the system has both fixed loads and price
sensitive loads. A 30 bus system is used for this illustration. The bids and offers parameters of the
system used here are given Table 1. There are some loads which do not submit offer function implying
that they are ready to accept power at any cost. In this system the total fixed demand by those
consumers is 75.4 MW. MCP of the system is obtained at the intersection point of aggregate demand
and supply curves as shown in figure 6.
Gen No.
1
2
22
27
23
13
148
ai
0.02
0.0175
0.0625
0.0083
0.025
0.025
Table 1. The Bids of 30 Bus System
bi
Dem No.
αi
2.00
2
-0.02
1.75
7
-0.02
1.00
8
-0.02
3.25
12
-0.02
3.00
21
-0.02
3.00
30
-0.02
βi
6.0
4.4
4.8
4.2
4.8
4.0
Vol. 1,Issue 2,pp.144-150
International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
Figure 6. Demand and Generation vs Incremental Cost
The final clearing price is: λ= 3.861
The schedules obtained as per the clearing price have been shown in table 2.
Gen No.
1
2
22
27
23
13
∑Gen.
Table 2. The Schedules Obtained For 30 Bus System
Pgi
Dem No.
Pdi
46.550
2
53.45
60.343
7
13.45
22.8960
8
23.45
36.867
12
8.45
17.2400
21
23.45
17.2400
30
3.45
200.9745
∑Var.Dem
125.85
F. Case 6: Mixed Load with Limit Violation
Consider the system as in case 5, and suppose that in addition there is a maximum limit of 50 MW on
the demand of consumer 2, Pd2max = 50MW.
Figure 7. Demand and Generation vs Incremental Cost
The final clearing price is : λ = 3.85 Rs/MW
The schedules obtained as per the new clearing price are:
Pg1 = 46.25MW, Pg2 = 60.00MW, Pg22 = 22.8MW, Pg27 = 36.1446MW, Pg23 = 17.00MW,
Pg13 = 17.00MW
Pd2 = 50.00MW, Pd7 = 13.75MW, Pd8 = 23.75MW, Pd12 = 8.75MW, Pd21 = 23.75MW,
Pd30 = 3.75MW
149
Vol. 1,Issue 2,pp.144-150
International Journal of Advances in Engineering & Technology, May 2011.
©IJAET
ISSN: 2231-1963
4. CONCLUSION
A graphical scheme for determining the market clearing price using quadratic bid functions is
proposed in this paper. The suggested algorithm uses a closed loop solution for the calculation of the
incremental cost (λ) which is used iteratively for the calculation of the clearing price and the schedules
corresponding to each generator and consumer bid.
The results obtained from the proposed algorithm are congruent with those obtained in [2] as shown in
Table 3. Using the iterative approach the need for calculating the intermediate values of incremental
cost has been eliminated hence increasing the system efficiency and reducing the computation time to
a large extent.
Table 3: Comparison of the results
Case no.
Clearing Price as per [2]
1
2
3
4
5
6
4.6792
4.735
4.5584
4.6375
3.862
3.8499
Clearing Price as
proposed Algorithm
4.679
4.735
4.558
4.637
3.862
3.85
per
REFERENCES
[1]
N. Fabra N.H. von der Fehr and D. Harbord,(August,2002), “Modeling Electricity Auctions”, The
Electricity Journal. Elsevier Science, Inc.pp. 72 - 81
[2]
Bijuna Kunju K and P S Nagendra Rao, (2008) “Determination of market clearing price in pool markets
with elastic Demand.” Fifteenth National Power Systems Conference (NPSC), IIT Bombay.
[3]
S. Stoft, Power System Economics: Designing Markets For Electricity. IEEE Press: John Wiley
Sons, 2002.
[4]
G. Gutiérrez, J. Quiñónez and G. B. Sheblé, “Market Clearing Price Discovery in a Single and DoubleSide Auction Market Mechanisms: Linear Programming Solution”
[5]
J. Alonso, A. Trias, V. Gaitan, J. J.Alba,(1999) “Thermal plant bids and market clearing in an electricity
pool. Minimization of costs vs. minimization of consumer payments.” IEEE Transactions on Power
Systems, Vol. 14, No. 4.
[6]
Mary B. Cain and Fernando L. Alvarado, (2004) “Electricity Market Studies: Linear Versus Quadratic
Costs” IEEE
and
Authors
Tanay Joshi is an undergraduate student in school of electrical engineering at VIT University,
Vellore, TamilNadu doing B.Tech in Electrical and Electronics Engineering. His areas of
interest are Energy Markets, Demand Response Management and smart energy management
systems for homes and industries.
Gagan Uberoi is an undergraduate student in school of electrical engineering at VIT
University, Vellore, Tamil Nadu. His areas of interest are Deregulation of power system
networks, energy management systems and energy efficient buildings and green building
technology.
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